How can we find the value function by solving a system of linear equations without knowing the policy?
Your equations all look correct to me.
It is not possible to solve the linear equation for state values in the vector $V$ without knowing the policy.
There are ways of working with MDPs, through sampling of actions, state transitions and rewards, where it is possible to estimate value functions without knowing either $\pi(a|s)$ or $P^{a}_{ss'}$. For instance, Monte Carlo policy evaluation or single-step TD learning can both do this. It is also common to work with $\pi(a|s)$ known but $P^{a}_{ss'}$ and $R^{a}_{s}$ unknown in model-free control algoithms such as Q learning.
However, in your case, you are correct, in order to resolve the simultaneous equations you have presented, you do need to know $\pi(a|s)$
This is not as limiting as you might think. You can construct a control method using simultaneous equations, by starting with the policy set to some arbitrary policy. Either a randomly-chosen deterministic policy or the equiprobable policy are reasonable first guesses. Then, after each solution to linear equations, you improve the policy so that each action choice maximises the expected return. This is essentially the policy iteration algorithm but replacing the policy evaluation step with the linear equations method for calculating the values.